So, my cat gave birth in a closet probably 6 or 7 hours ago.
The problem is, I'm supposed to be moving from one house to another before the 15th.

My questions :
When is it okay for me to move the kitten, what can I do to make sure that it won't get hurt when Im moving it and how can I make sure that the mother doesn't freak out when I try to move it?

You shouldnt have much of a problem moving them, if the mother trust you enough she will let you move the kittens, my cat let us move her and the kittens to a new room after 15 days of them being born, just put them all in a nice dark and quiet place when you move them, keep mommy with them at all times. You should be ok in my opinion, as for the other thing, idk, try googling it to see.